Disjoint Families Research Articles

Interaction Dynamics of Gap Solitons in Dual-Core Bragg Gratings With Cubic-Quintic Nonlinearity

Interactions between quiescent gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity are systematically investigated. In a previous work, it was found that the model supports symmetric and asymmetric soliton solutions. For each of these categories, there exist two disjoint families of quiescent gap solitons. One family can be regarded as the generalization of the quiescent gap solitons in dual-core Bragg gratings with Kerr nonlinearity (called Type 1). On the other hand, the other family is found in the region, where the quintic nonlinearity is dominant (called Type 2). The interactions of in-phase Type 1 asymmetric solitons result in a range of outcomes, namely, fusion into a single zero-velocity soliton, asymmetrical separation of solitons, symmetrical separation of solitons, formation of three solitons, and the destruction of solitons. In the case of symmetric solitons, interactions of in-phase Type 1 solitons may lead to fusion in the form of a quiescent soliton, asymmetrically separating solitons, or the formation of three solitons. It is also found that the interaction of in-phase asymmetric Type 2 solitons results in their destruction. Also, the effects of the quintic nonlinearity, coupling coefficients, initial separation, and initial phase difference on the outcomes of the interactions are analyzed. Also, in the case of asymmetric solitons, the outcomes of the soliton-soliton interactions for specular-symmetric and cross-symmetric initial configurations are compared.

The Rothberger property on $C_p(\Psi(\mathcal A),2)$

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle\,\mathcal U_n:n\in \omega\,\rangle$ of open covers of $X$, there exists $U_n\in \mathcal U_n$ for each $n\in\omega$ such that $X = \bigcup_{n\in \omega}U_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p(\Psi(\mathcal A),2)^n$ to have the Rothberger property when $\mathcal A$ is a Mrowka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $\mathcal A$ for which the space $C_p(\Psi(\mathcal A),2)^n\,$ is Rothberger for all $n\in\omega$.

Rotating neutron stars with exotic cores: masses, radii, stability

A set of theoretical mass-radius relations for rigidly rotating neutron stars with exotic cores, obtained in various theories of dense matter, is reviewed. Two basic observational constraints are used: the largest measured rotation frequency (716 Hz) and the maximum measured mass ($2\;M_\odot$). Present status of measuring the radii of neutron stars is described. The theory of rigidly rotating stars in general relativity is reviewed and limitations of the slow rotation approximation are pointed out. Mass-radius relations for rotating neutron stars with hyperon and quark cores are illustrated using several models. Problems related to the non-uniqueness of the crust-core matching are mentioned. Limits on rigid rotation resulting from the mass-shedding instability and the instability with respect to the axisymmetric perturbations are summarized. The problem of instabilities and of the back-bending phenomenon are discussed in detail. Metastability and instability of a neutron star core in the case of a first-order phase transition, both between pure phases, and into a mixed-phase state, are reviewed. The case of two disjoint families (branches) of rotating neutron stars is discussed and generic features of neutron-star families and of core-quakes triggered by the instabilities are considered.

More Specific Signal Detection in Functional Magnetic Resonance Imaging by False Discovery Rate Control for Hierarchically Structured Systems of Hypotheses.

Signal detection in functional magnetic resonance imaging (fMRI) inherently involves the problem of testing a large number of hypotheses. A popular strategy to address this multiplicity is the control of the false discovery rate (FDR). In this work we consider the case where prior knowledge is available to partition the set of all hypotheses into disjoint subsets or families, e. g., by a-priori knowledge on the functionality of certain regions of interest. If the proportion of true null hypotheses differs between families, this structural information can be used to increase statistical power. We propose a two-stage multiple test procedure which first excludes those families from the analysis for which there is no strong evidence for containing true alternatives. We show control of the family-wise error rate at this first stage of testing. Then, at the second stage, we proceed to test the hypotheses within each non-excluded family and obtain asymptotic control of the FDR within each family at this second stage. Our main mathematical result is that this two-stage strategy implies asymptotic control of the FDR with respect to all hypotheses. In simulations we demonstrate the increased power of this new procedure in comparison with established procedures in situations with highly unbalanced families. Finally, we apply the proposed method to simulated and to real fMRI data.

Fibers of continuous real-valued functions on ψ-spaces

We consider continuous real-valued functions with domain either a ψ-space (studied by S. Mrowka, J. Isbell, and others) or a generalized ψ-space introduced by A. Dow and J. Vaughan. A cardinal κ≥ω is called a rich cardinal provided for every infinite, maximal almost disjoint family M of countably infinite subsets of κ (MADF) and for every continuous f:ψ(κ,M)→R defined on the associated space ψ=ψ(κ.M), there exists r∈R such that |f−1(r)|=|ψ|. Dow and Vaughan proved that ω is a rich cardinal if and only if a=c, where a is the smallest cardinality of a MADF on ω. We prove that a=c if and only if, for all ω≤κ≤c, κ is a rich cardinal, if and only if for every n<ω, ωn is a rich cardinal. We prove every κ>c is rich using a set-theoretic hypothesis weaker than GCH.

The Rothberger property on [formula omitted

In this paper we analyze the Rothberger property on Cp(X,2). A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 〈Un:n∈ω〉 of open covers of X, there exists Un∈Un for each n∈ω such that X=⋃n∈ωUn. We show the following: (1) if Cp(X,2) is Rothberger, then X is pseudocompact; (2) for every pseudocompact Sokolov space X with t⁎(X)≤ω, Cp(X,2) is Rothberger; and (3) assuming CH (the continuum hypothesis) there is a maximal almost disjoint family A for which the space Cp(Ψ(A),2) is Rothberger. Moreover, we characterize the Rothberger property on Cp(L,2) when L is a GO-space.

Maximal cofinitary groups revisited

Let κ be an arbitrary regular infinite cardinal and let denote the set of κ‐maximal cofinitary groups. We show that if holds and C is a closed set of cardinals such that , , if then , , then there is a generic extension in which cofinalities have not been changed and such that . The theorem generalizes a result of Brendle, Spinas and Zhang (cf. ) regarding the possible sizes of maximal cofinitary groups. Our techniques easily modify to provide analogous results for the spectra of maximal κ‐almost disjoint families in , maximal families of κ‐almost disjoint permutations on κ and maximal families of κ‐almost disjoint functions in . In addition we construct a κ‐Cohen indestructible κ‐maximal cofinitary group and so establish the consistency of , which for is due to Yi Zhang (cf. ).

MAD SPECTRA

AbstractThe mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets ${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption $\vartheta &amp;#x5e;{ &amp;#x3c; \vartheta } &amp;#x3d; \vartheta $, where $\vartheta &amp;#x3d; {\rm{min}}\left( {\rm {\cal A}} \right) $.

Extending the class of known Stone–Čech remainders for ψ-spaces

We are looking at the Stone–Čech remainders of ψ-spaces with maximal almost disjoint families taken from [κ]ω, for κ≥ω. Such spaces include all standard ψ-space where κ=ω. Well-known Stone–Čech remainders of ψ-spaces of this type are compact ordinals λ for λ<c+⋅ω, A. Dow and J.E. Vaughan [2], compact metric spaces and the larger class of compact first countable separable spaces, Jun Terasawa [4]. We give a class of spaces A that includes the aforementioned spaces, where for each X∈A, we have a single construction that builds a ψ-space, ψ(M), where β(X)=β(ψ(M))∖ψ(M). Clearly for such an A if X is compact and X∈A, then X=β(ψ(M))∖ψ(M). In addition to A containing in the Stone–Čech compactification of its members the compact ordinals λ<c+⋅ω, compact metric spaces and compact separable first countable spaces, we will also have as the Stone–Čech compactifications of members of A the following: the Stone–Čech compactification of completely regular separable countably compact Fréchet spaces; under the product topology for κ≤c, 2κ, and more generally Mκ where M is a compact metric space; closed intervals of any first countable connected totally ordered space of density less than or equal to c. It is consistent that we will also have closed intervals of any Suslin line. We define A to be the class of spaces X where X is a completely regular countably compact Fréchet space, with a dense subset D, so that for each x∈X, there exists a neighborhood Ux of x, with |Ux∩D|≤c.

Regular [formula omitted]-diagonal and mappings

In this paper, we prove that an example by Mysior has no regular Gδ-diagonal, which can be used to show that open ≤2-to-one maps need not preserve regular Gδ-diagonals or zero-set diagonals. We also discuss the diagonal of the Mrówka space. Let A be a maximal almost disjoint family on N, then for any countable subset B of A, the Mrówka space ψ(A∖B) has no regular Gδ-diagonal.

On Keller’s Conjecture in Dimension Seven

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. Let $r^-(T)=\min_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$ and $r^+(T)=\max_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$. It is known that if $r^-(T)\leq 2$ or $r^+(T)\geq 6$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $r^+(T)=5$. Thus, if $[0,1)^d+T$ is a counterexample to Keller's conjecture in dimension seven, then $r^-(T),r^+(T)\in \{3,4\}$.

Template iterations and maximal cofinitary groups

In [2], Jorg Brendle used Hechler’s forcing notion for adding a maximal almost family along an appropriate template forcing construction to show that a (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that ag, the minimal size of maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing construction. Additionally we obtain that ap, the minimal size of a maximal family of almost disjoint permutations, and ae, the minimal size of a maximal eventually different family, can be of countable cofinality.

Stability of gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity

We investigate the existence and stability of quiescent gap solitons in a system of two linearly coupled Bragg gratings with cubic-quintic nonlinearity. It is found that the model supports two disjoint families of solitons that fill the entire bandgap. There exist symmetric and asymmetric solitons in each family. Exact analytical solutions of symmetric solitons for both families are found. The stability of solitons is investigated by means of systematic numerical stability analysis. The effect of quintic nonlinearity and coupling coefficient on the stability of solitons and the stability regions are analyzed.

Splitting Families and Complete Separability

AbstractWe answer a question from Raghavan and Steprans by showing that Then we use this to construct a completely separable maximal almost disjoint family under a, partially answering a question of Shelah.

Monge problem in metric measure spaces with Riemannian curvature-dimension condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N<∞. For the first marginal measure, we assume that μ0≪m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value.Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Mackey hyperplanes/enlargements for Tweddle’s space

Tweddle (India J Math 50:99–114, 2008) showed that Banach’s bounded sequence space retains certain properties when endowed with a topology \(\eta \) defined by a maximal almost disjoint (mad) family of natural numbers. In particular, it is Mackey. But it lacks most weak barrelledness properties, forcing Saxon and Tweddle (Adv Math 145:230–238, 1999) to find elsewhere the first example of a Mackey \(\aleph _{0}\)-barrelled space that is not barrelled. We shall show that \(\eta \) varies in tightness as the size of its defining mad family: Tweddle’s spaces are not all isomorphic. But their countable-codimensional subspaces and countable enlargements are all Mackey. Likewise for the Saxon and Tweddle example, which multiplies our scant supply of such. Independently interesting methods are developed.

(a)-spaces and selectively (a)-spaces from almost disjoint families

We investigate a selective version of property (a) and prove a number of results showing that, under certain set theoretical conditions, (a) spaces and selectively (a) spaces behave in a very similar way, at least for separable spaces. Several results regarding the presence of the referred selective version in spaces from almost disjoint families are established; in particular, we give a combinatorial characterization of such presence. Consistent set theoretical hypotheses implying equivalence between being (a) and being selectively (a) within the referred class are presented, as well as hypotheses implying non-equivalence. We also show that the Continuum Hypothesis is independent of the statement asserting the above mentioned equivalence. The paper finishes by presenting some notes and questions on the role of set theoretical assumptions in the subject.

Moving Bragg grating solitons in a cubic-quintic nonlinear medium with dispersive reflectivity

The stability and collision dynamics of moving solitons in Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity are investigated. Two disjoint families of solitons are found on the plane of the coefficient of quintic nonlinearity versus the normalized frequency (η,Ω(norm)). Through numerical stability analysis, we have identified stability regions on the (η,Ω(norm)) plane for various values of dispersive reflectivity parameter (m) and velocity (v). The size of stability regions is found to be dependent on m and v. Collisions of counterpropgating Type 1 and Type 2 solitons have been systematically investigated. It is found that for low to moderate values of dispersive reflectivity, the collisions of Type 1 solitons can result in various outcomes such as separation of solitons with reduced, increased, unchanged, or asymmetric velocities and generation of a quiescent soliton by merger or formation of three solitons. For strong dispersive reflectivity (e.g., m=0.5), the collisions of low-velocity in-phase Type 1 solitons may lead to repulsion of solitons, asymmetric separation, merger into a single soliton, or formation of three solitons (one quiescent and two moving solitons). At higher velocities collisions predominantly lead to the formation of three solitons. For m=0.5, in-phase Type 2 solitons may repel or form a temporary bound state of quiescent Type 1 solitons that subsequently splits into two asymmetrically separating Type 1 solitons. π-out-of-phase Type 2 solitons may also merge to form a quiescent Type 1 soliton.

Stability and collisions of moving Bragg grating solitons in a cubic-quintic nonlinear medium

The existence, stability, and collisions of moving solitons in Bragg gratings (BGs) in a cubic-quintic nonlinear medium are investigated. Two disjoint families of solitons that are separated by a border are identified. One family (Type 1) can be considered as the generalization of the moving solitons in BGs written in a cubic nonlinear medium. The other family (Type 2) occurs in regions where quintic nonlinearity dominates. Through systematic numerical stability analysis, the stability regions in the plane of quintic nonlinearity versus frequency have been determined. It is found that the stability regions are dependent on the velocity of solitons. The collisions of counterpropagating solitons have been systematically investigated. Collisions of in-phase Type 1 solitons can result in a variety of outcomes including forming two asymmetrically separating solitons and passing through each other and separating symmetrically with reduced, unchanged, or increased velocities. In certain parameter regions, solitons merge to form a quiescent one. An outcome that has not been reported previously for uniform gratings is the formation of a quiescent soliton and two symmetrically separating solitons. This outcome is found to be more robust than the merger.

Conditions for families of disjoint [formula omitted]-connected subgraphs in a graph

In [T. Böhme A. Kostochka, Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density, Discrete Math. 309 (4) (2009) 997–1000.], Böhme and Kostochka showed that every large enough graph with sufficient edge density contains either a k-connected subgraph of order at least r or a family of r vertex-disjoint k-connected subgraphs. Motivated by this, in this note we explore the latter conclusion of their work and give conditions that ensure a graph G contains a family of vertex-disjoint k-connected subgraphs. In particular, we show that a graph of order n with at least 223ksn120ϵ edges contains a family of s disjoint k-connected subgraphs each of order at most ϵn. We also show for k≥2, the vertices of a graph with minimum degree at least 2kn can be partitioned into k-connected subgraphs. The degree condition in the latter result is asymptotically the best possible as a function of n.